| L(s) = 1 | + 1.46·2-s − 3-s + 0.156·4-s − 5-s − 1.46·6-s − 1.71·7-s − 2.70·8-s + 9-s − 1.46·10-s + 0.820·11-s − 0.156·12-s + 3.73·13-s − 2.51·14-s + 15-s − 4.28·16-s + 1.94·17-s + 1.46·18-s + 6.67·19-s − 0.156·20-s + 1.71·21-s + 1.20·22-s + 3.67·23-s + 2.70·24-s + 25-s + 5.49·26-s − 27-s − 0.268·28-s + ⋯ |
| L(s) = 1 | + 1.03·2-s − 0.577·3-s + 0.0784·4-s − 0.447·5-s − 0.599·6-s − 0.647·7-s − 0.956·8-s + 0.333·9-s − 0.464·10-s + 0.247·11-s − 0.0453·12-s + 1.03·13-s − 0.671·14-s + 0.258·15-s − 1.07·16-s + 0.471·17-s + 0.346·18-s + 1.53·19-s − 0.0351·20-s + 0.373·21-s + 0.256·22-s + 0.766·23-s + 0.552·24-s + 0.200·25-s + 1.07·26-s − 0.192·27-s − 0.0507·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.795529969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.795529969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 1.46T + 2T^{2} \) |
| 7 | \( 1 + 1.71T + 7T^{2} \) |
| 11 | \( 1 - 0.820T + 11T^{2} \) |
| 13 | \( 1 - 3.73T + 13T^{2} \) |
| 17 | \( 1 - 1.94T + 17T^{2} \) |
| 19 | \( 1 - 6.67T + 19T^{2} \) |
| 23 | \( 1 - 3.67T + 23T^{2} \) |
| 29 | \( 1 - 1.77T + 29T^{2} \) |
| 31 | \( 1 + 0.868T + 31T^{2} \) |
| 37 | \( 1 - 7.37T + 37T^{2} \) |
| 41 | \( 1 + 4.03T + 41T^{2} \) |
| 43 | \( 1 - 9.34T + 43T^{2} \) |
| 47 | \( 1 - 3.27T + 47T^{2} \) |
| 53 | \( 1 + 3.15T + 53T^{2} \) |
| 59 | \( 1 - 2.44T + 59T^{2} \) |
| 61 | \( 1 + 6.78T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 9.85T + 73T^{2} \) |
| 83 | \( 1 - 8.82T + 83T^{2} \) |
| 89 | \( 1 + 15.3T + 89T^{2} \) |
| 97 | \( 1 - 2.70T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.677030885747720379016372401625, −9.128276948251119464153932599352, −8.013510828161022166247053678364, −6.99461078196191334255014149665, −6.15981521911767337587120327388, −5.50908954422069914598195470309, −4.59549760181203944146507484228, −3.67119884588860290266017233541, −3.00512468021345711473371556685, −0.914577737991227351052994010209,
0.914577737991227351052994010209, 3.00512468021345711473371556685, 3.67119884588860290266017233541, 4.59549760181203944146507484228, 5.50908954422069914598195470309, 6.15981521911767337587120327388, 6.99461078196191334255014149665, 8.013510828161022166247053678364, 9.128276948251119464153932599352, 9.677030885747720379016372401625
